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2025-03-24T23:47:52Z

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In [[mathematics]], the '''incomplete polylogarithm''' function is related to the [[polylogarithm]] function. It is sometimes known as the [[incomplete Fermi–Dirac integral]] or the incomplete [[Bose–Einstein statistics|Bose–Einstein]] integral. It may be defined by:

:<math>
\operatorname{Li}_s(b,z) = \frac{1}{\Gamma(s)}\int_b^\infty \frac{x^{s-1}}{e^{x}/z-1}~dx.
</math>

Expanding about z=0 and integrating gives a series representation:

:<math>
\operatorname{Li}_s(b,z) = \sum_{k=1}^\infty \frac{z^k}{k^s}~\frac{\Gamma(s,kb)}{\Gamma(s)}
</math>

where Γ(s) is the [[gamma function]] and Γ(s,x) is the upper [[incomplete gamma function]]. Since Γ(s,0)=Γ(s), it follows that:

:<math>
\operatorname{Li}_s(0,z) =\operatorname{Li}_s(z)
</math>

where Li<sub>s</sub>(.) is the polylogarithm function.

== References ==
* GNU Scientific Library - Reference Manual https://www.gnu.org/software/gsl/manual/gsl-ref.html#SEC117

[[Category:Special functions]]

Incomplete polylogarithm - Revision history

imported>Sleigh: sp